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平行应急疏散系统:基本概念、体系框架及其应用

周敏 董海荣 徐惠春 李浥东 王飞跃

吴小雪, 丁大伟, 任莹莹, 刘贺平.二维FM系统的同时故障检测与控制.自动化学报, 2021, 47(1): 224-234 doi: 10.16383/j.aas.c180673
引用本文: 周敏, 董海荣, 徐惠春, 李浥东, 王飞跃. 平行应急疏散系统:基本概念、体系框架及其应用. 自动化学报, 2019, 45(6): 1074-1086. doi: 10.16383/j.aas.c180558
Wu Xiao-Xue, Ding Da-Wei, Ren Ying-Ying, Liu He-Ping. Simultaneous fault detection and control of two-dimensional Fornasini-Marchesini systems. Acta Automatica Sinica, 2021, 47(1): 224-234 doi: 10.16383/j.aas.c180673
Citation: ZHOU Min, DONG Hai-Rong, XU Hui-Chun, LI Yi-Dong, WANG Fei-Yue. Parallel Emergency Evacuation Systems: Basic Concept, Framework and Applications. ACTA AUTOMATICA SINICA, 2019, 45(6): 1074-1086. doi: 10.16383/j.aas.c180558

平行应急疏散系统:基本概念、体系框架及其应用

doi: 10.16383/j.aas.c180558
基金项目: 

博士后创新人才支持计划 BX20190029

中央高校基本科研业务费 2018JBZ002

中央高校基本科研业务费 2019JBM079

详细信息
    作者简介:

    董海荣  北京交通大学轨道交通控制与安全国家重点实验室教授.主要研究方向为列车运行控制与辅助驾驶, 行人动力学.E-mail:hrdong@bjtu.edu.cn

    徐惠春  中国铁路北京局集团有限公司科技和信息化部主任, 教授级高工, 中国铁路总公司百千万人才专业带头人.主要研究方向为铁路基建项目工程建设, 铁路信息系统科技研发, 铁路技术规章与标准制定.E-mail:xhc6789@139.com

    李浥东  北京交通大学计算机与信息技术学院副教授.主要研究方向为多媒体计算, 数据挖掘, 云计算与高性能计算和智能交通系统.E-mail:ydli@bjtu.edu.cn

    王飞跃  中国科学院自动化研究所复杂系统管理与控制国家重点实验室主任, 国防科技大学军事计算实验与平行系统技术研究中心主任, 中国科学院大学中国经济与社会安全研究中心主任, 青岛智能产业技术研究院院长.主要研究方向为平行系统的方法与应用, 社会计算, 平行智能以及知识自动化.E-mail:feiyue.wang@ia.ac.cn

    通讯作者:

    周敏  北京交通大学轨道交通控制与安全国家重点实验室博士后.2019年获得北京交通大学交通信息工程及控制博士学位.主要研究方向为行人与疏散动力学和平行管理.本文通信作者.E-mail:zhoumin@bjtu.edu.cn

Parallel Emergency Evacuation Systems: Basic Concept, Framework and Applications

Funds: 

Postdoctoral Innovative Talent Project BX20190029

Fundamental Research Funds for Central Universities 2018JBZ002

Fundamental Research Funds for Central Universities 2019JBM079

More Information
    Author Bio:

    Professor at the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University. Her research interest covers train operation control and advanced driver assistance systems and pedestrian dynamics

    Professor of engineering and the director in the Science and Information Department, China Railway Beijing Group Co., Ltd., Professional leader of Millions of Talents at China Railway Corporation. His research interest covers railway infrastructure engineering, science and information technologies and technical regulations and standards

    Associate professor at the School of Computer and Information Technology, Beijing Jiaotong University. His research interest covers multimedia computing, data mining, cloud computing and high performance computing, and intelligent transportation systems

    State specially appointed expert and director of the State Key Laboratory for Management and Control of Complex Systems, Institute of Automation, Chinese Academy of Sciences. Professor of the Research Center for Computational Experiments and Parallel Systems Technology, National University of Defense Technology. Director of China Economic and Social Security Research Center in University of Chinese Academy of Sciences. Dean of Qingdao Academy of Intelligent Industries. His research interest covers methods and applications for parallel systems, social computing, parallel intelligence, and knowledge automation

    Corresponding author: ZHOU Min Postdoctoral research fellow at the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University. He received his Ph. D. degree in traffic information engineering and control from Beijing Jiaotong University in 2019. His research interest covers pedestrian and evacuation dynamics and parallel management. Corresponding author of this paper
  • 摘要: 突发事件通常具有难以预测、多成因关联、危害性大及演变复杂等特点,应急情况下如何安全高效疏散人员是应急管控领域的重要研究内容.本文将基于人工系统(Artificial systems,A)、计算实验(Computational experiments,C)、平行执行(Parallel excution,P)方法的平行系统理论引入到应急管控领域,提出平行应急疏散系统(Parallel emergency evacuation systems,PeES)基本概念,构建系统体系框架及集成平台,并介绍人工应急疏散系统、计算实验、平行执行等主要功能模块的基本功能及实现方法.通过PeES能实现虚实应急疏散系统的管理与控制、应急方案的实验与评估以及相关人员的学习与培训.最后,以轨道交通枢纽站火灾场景下的乘客应急疏散为典型应用对平行应急疏散系统进行初步验证.
  • 二维系统与现代过程控制密切相关, 在污水处理、多维数字滤波器、卫星气象云图分析、图像处理等领域有广泛的应用[1-2].由于二维系统具有丰富的工程物理背景, 至今仍是控制领域的研究热点之一.近年来二维系统的分析和控制器设计方面取得了诸多成果, 如文献[3]给出了二维离散线性模型的稳定性判据, 文献[4]给出了二维离散系统的状态反馈控制器设计方法.在此基础上, 二维系统的滤波器设计、$ {H_\infty} $控制等问题也得到了深入研究[5-6].

    另一方面, 现代工业过程对于系统安全性和可靠性的要求日益提高, 因此故障检测与故障诊断问题越来越受到重视[7-10].与一维系统相比, 二维系统由于结构复杂, 其故障检测滤波器/观测器的设计难度更大, 需要提出新技术解决设计过程中遇到的非凸问题.此外, 二维系统的残差评价函数以及阈值的设计亦不同于一维系统, 需根据二维系统的特点构造合适的残差评价函数及阈值.因此, 研究二维系统的故障检测问题是非常必要的, 也是存在挑战的.然而, 现有文献的研究成果相对较少, 其中文献[11-13]研究了二维系统的故障检测问题, 采用了全频设计方法, 即没有考虑故障发生的有限频信息; 文献[14]研究了二维Roesser系统在传感器失效情况下的故障检测问题; 文献[15]研究了二维FM系统的故障检测问题.

    文献[16]指出, 系统设计时控制单元和故障检测单元是相互关联和作用的, 如果将控制单元和故障检测单元分开设计, 容易忽视两个单元间的关联性:设计控制单元时容易影响故障检测效果, 而设计故障检测单元时可能会影响控制效果.解决该问题的方法之一是采用集成设计策略, 一方面可减少设计步骤, 一定程度上降低设计的复杂度[17]; 另一方面, 集成设计可以考虑控制单元和故障检测单元的关联性和相互作用, 即兼顾和平衡控制性能和故障检测性能.此外, 集成设计也有助于在控制器中结合容错特性[18].目前, 集成设计的研究成果主要集中在一维系统[19], 二维系统的相关研究成果较少[20]. FM模型作为一类重要的二维系统模型, 其同时故障检测与控制方法具有一定的理论意义和实际应用价值, 但该研究未见相关报道, 这是本文的研究动机之一.

    本文研究二维FM系统的同时故障检测与控制问题, 采用有限频性能指标刻画故障和干扰信号的有限频特性, 提出构造切平面方法和两步算法来解决设计过程中出现的非凸问题.本文所设计的故障检测滤波器/控制器可以同时实现控制功能和故障检测功能.此外, 以往研究成果常采用递增的残差评价函数, 故障被排除后容易产生故障误报.本文针对二维系统的故障检测问题, 采用新的残差评价函数, 可降低故障误报率.

    考虑如下二维离散FM模型[21]:

    $$ \begin{align} {\pmb x}({ i} + 1, { j} + 1) = \, & A_1{\pmb x}({ i}, { j} + 1) + A_2{\pmb x}({ i} + 1, { j}) +\\ & B_{{ d}1}{\pmb d}({ i}, { j} + 1) + B_{{ d}2}{\pmb d}({ i} + 1, { j}) +\\ & B_{{ u}1}{\pmb u}({ i}, j + 1) + B_{u2}{\pmb u}({ i}+1, j) +\\ & B_{{ f}1}{\pmb f}({ i}, j + 1) + B_{{ f}2}{\pmb f}({ i} + 1, { j}) \\ \pmb y({ i}, { j}) = \, &C{\pmb x}({ i}, { j}) + D_{ d}{\pmb d}({ i}, { j}) + D_{ f}{\pmb f}({ i}, { j}) \\ \pmb z({ i}, { j}) = \, &E{\pmb x}({ i}, { j}) +F_{ u} {\pmb u}({ i}, { j}) \end{align} $$ (1)

    其中, $ \pmb x(i, j) \in {\bf R}^n $为状态向量, $ \pmb d(i, j) \in {\bf R}^{{n_d}} $为外部扰动, $ \pmb f(i, j) \in {\bf R}^{{n_f}} $为故障信号, $ \pmb y(i, j) \in {\bf R}^{{n_y}} $为测量输出, $ \pmb z(i, j)\in {\bf R}^{{n_z}} $为被控输出, $ {A_1} $, $ {A_2} $, $ {B_{u1}} $, $ {B_{u2}} $, $ {B_{f1}} $, $ {B_{f2}} $, $ {B_{d1}} $, $ {B_{d2}} $, $ C $, $ {D_d} $, $ {D_f} $, $ E $, $ {F_u} $为已知的具有适当维数的常数矩阵.

    注1.  对于FM模型, 由于状态向量$ \pmb x(i+1, j+1) $可以看做$ i $方向上对$ \pmb x(i, j+1) $或者是在$ j $方向上对$ \pmb x(i, j+1) $进行的一步前移运算, 故FM模型需要三个向量来描述[22].

    本文目标是构造如下形式的故障检测滤波器/控制器:

    $$ \begin{align} \hat{\pmb x}(i + 1, j + 1) = \, &{\hat A_1} \hat{\pmb x}(i, j + 1) + {\hat A_2} {\hat{\pmb x}}(i + 1, j) +\\ & {\hat B_1}\pmb y(i, j + 1) + {\hat B_2}\pmb y(i + 1, j) \\ \pmb u(i, j) = \, & {\hat C_c} \hat{\pmb x}(i, j) \\ \hat{\pmb y}(i, j) = \, & {\hat C_0} \hat{\pmb x}(i, j) \end{align} $$ (2)

    其中, $ \hat{\pmb x}(i, j) \in {\bf R}^n $为状态估计, $ \pmb u(i, j) \in{\bf R}^{{n_u}} $为控制输入, $ \hat{\pmb y}(i, j) \in {\bf R}^{{n_y}} $是对输出的估计, $ {\hat A_1} $, $ {\hat A_2} $, $ {\hat B_1} $, $ {\hat B_2} $, $ {\hat C_0} $, $ {\hat C_c} $是待定的滤波器和控制器参数.

    定义$ \bar{\pmb x}(i, j) = \left[{\begin{array}{*{20}{c}} {{\pmb x^{\rm T}(i, j)}}&{ \hat{\pmb x}^{\rm T}(i, j)}\end{array}}\right] ^{\rm T} $, 及残差信号$ \tilde{\pmb y}(i, j) = \pmb y(i, j) - \hat{\pmb y}(i, j), $结合式(1)和式(2)可得增广系统:

    $$ \begin{align} \bar{\pmb x}(i + 1, j + 1) = \, & {\bar A_1}\bar{\pmb x}(i, j + 1) + {\bar A_2}\bar{\pmb x}(i+ 1, j)+\\ & {\bar B_{d1}}\pmb d(i, j + 1)+ {\bar B_{d2}}\pmb d(i + 1, j) +\\ & {\bar B_{f1}}\pmb f(i, j + 1)+ {\bar B_{f2}}\pmb f(i + 1, j)\\ \tilde{\pmb y}(i, j) = \, & \tilde C\bar{\pmb x}(i, j) + {\tilde D_d}\pmb d(i, j) + {\tilde D_f}\pmb f(i, j)\\ {\pmb z}(i, j) = \, & \tilde E \bar{\pmb x}(i, j) \end{align} $$ (3)

    其中

    $$ \begin{align} \, &{\bar A_1} = \left[ {\begin{array}{*{20}{c}} {{A_1}}&{{B_{u1}}{{\hat C}_c}}\\ {{{\hat B}_1}C}&{{{\hat A}_1}} \end{array}} \right], \; {\bar A_2} = \left[ {\begin{array}{*{20}{c}} {{A_2}}&{{B_{u2}}{{\hat C}_c}}\\ {{{\hat B}_2}C}&{{{\hat A}_2}} \end{array}} \right] \\ \, &{\bar B_{d1}} = \left[ {\begin{array}{*{20}{c}} {{B_d}_1}\\ {{{\hat B}_1}{D_d}} \end{array}} \right], \; {\bar B_{d2}} = \left[ {\begin{array}{*{20}{c}} {{B_d}_2}\\ {{{\hat B}_2}{D_d}} \end{array}} \right] \\ \, &{\bar B_{f1}} = \left[ {\begin{array}{*{20}{c}} {{B_{f1}}}\\ {{{\hat B}_1}{D_f}} \end{array}} \right], \; {\bar B_{f2}} = \left[ {\begin{array}{*{20}{c}} {{B_{f2}}}\\ {{{\hat B}_2}{D_f}} \end{array}} \right] \\ \, &\tilde C = [\begin{array}{*{20}{c}} C&{ - {{\hat C}_0}} \end{array}], \; {\tilde D_d} = {D_d} \\ \, &{\tilde D_f} = {D_f}, \; \tilde E = [\begin{array}{*{20}{c}} E&{{F_u}{{\hat C}_c}} \end{array}] \end{align} $$ (4)

    增广系统从故障$ \pmb f(i, j) $、干扰$ \pmb d(i, j) $到残差$ \tilde{\pmb y}(i, j) $和被控输出$ \pmb z(i, j) $的传递函数分别由下式给出:

    $$ \begin{align} {G_{\tilde yf}}({\omega _1}, {\omega _2}) = \, &\tilde C{({z_1}{z_2}I - {z_2}{\bar A_1} -{z_1}{\bar A_2})^{-1}}({z_2}{\bar B_{f1}} +\\ & {z_1}{\bar B_{f2}}) +{\tilde D_f} \end{align} $$ (5)
    $$ \begin{align} {G_{\tilde yd}}({\omega _1}, {\omega _2}) = \, &\tilde C{({z_1}{z_2}I - {z_2}{\bar A_1} - {z_1}{\bar A_2})^{ - 1}}({z_2}{\bar B_{d1}}+\\ & {z_1}{\bar B_{d2}}) + {\tilde D_d} \end{align} $$ (6)
    $$ \begin{align} {G_{zf}}({\omega _1}, {\omega _2}) = \, &\tilde E{({z_1}{z_2}I - {z_2}{\bar A_1} - {z_1}{\bar A_2})^{ - 1}}({z_2}{\bar B_{f1}}+ {z_1}{\bar B_{f2}}) \end{align} $$ (7)
    $$ \begin{align} {G_{zd}}({\omega _1}, {\omega _2}) = \, &\tilde E{({z_1}{z_2}I - {z_2}{\bar A_1} - {z_1}{\bar A_2})^{ - 1}}({z_2}{\bar B_{d1}}+ {z_1}{\bar B_{d2}}) \end{align} $$ (8)

    其中, $ {z_1} = {{\rm e}^{j\omega_1}} $, $ {z_2} = {{\rm e}^{j\omega_2}} $.

    本文要讨论的问题可归纳为:对于给定的二维FM系统, 设计故障检测滤波器/控制器(2), 使增广系统(3)渐近稳定, 同时满足如下控制指标和故障检测指标:

    $$ \begin{align} &\inf {\sigma _{\min }}({G_{\tilde yf}}({\omega _1}, {\omega _2})) > {\gamma _1}, \;\forall \left| {{\omega _1}} \right| \le {\bar \omega _{11}}, \left| {{\omega _2}} \right| \le {\bar \omega _{12}} \end{align} $$ (9)
    $$ \begin{align} &\sup {\sigma _{\max }}({G_{zf}}({\omega _1}, {\omega _2})) < {\beta _1}, \; \forall \left| {{\omega _1}} \right| \le {\bar \omega _{11}}, \left| {{\omega _2}} \right| \le {\bar \omega _{12}} \end{align} $$ (10)
    $$ \begin{align} &\sup {\sigma _{\max }}({G_{\tilde yd}}({\omega _1}, {\omega _2})) < {\gamma _2}, \;\forall \left| {{\omega _1}} \right| \le {\bar \omega _{21}}, \left| {{\omega _2}} \right| \le {\bar \omega _{22}} \end{align} $$ (11)
    $$ \begin{align} &\sup {\sigma _{\max }}({G_{zd}}({\omega _1}, {\omega _2})) < {\beta _2}, \;\forall \left| {{\omega _1}} \right| \le {\bar \omega _{21}}, \left| {{\omega _2}} \right| \le {\bar \omega _{22}} \end{align} $$ (12)

    这里, $ {\gamma _1} $, $ {\gamma _2} $, $ {\beta _1} $, $ {\beta _2} $是给定的正标量, $ {\bar \omega _{k1}} $, $ {\bar \omega _{k2}} \in \left[ {0, \pi } \right] $, $ k = 1, 2 $.

    注2.  有限频$ {H_ - } $指标(9)和有限频$ {H_\infty} $指标(10)$ \sim $ (12)是相应全频域指标的推广, 当$ {\bar \omega _{11}} = \bar \omega _{12} = \bar \omega _{21} = \bar \omega _{22} = \pi $时, 有限频性能指标退化为全频性能指标.

    注3.  式(9)和式(11)为故障检测性能指标, 这两个指标保证了发生在有限频域的故障对残差信号有足够大的影响, 同时外部干扰对残差信号的影响较小; 式(10)和式(12)为控制性能指标, 即抑制故障和干扰信号对被控输出的影响, 保证系统有一定的鲁棒性.

    本文需要用到如下引理:

    引理 1[23].  对于给定的对称矩阵$ \Psi $和矩阵$ \Gamma $, $ \Lambda $, 存在矩阵$ X $, 满足$ \Psi + \Gamma X{\Lambda ^{\rm T}} + \Lambda {X^{\rm T}}\Gamma < 0 $, 当且仅当以下等式成立:

    $$ \begin{align} {\Gamma ^ \bot }\Psi {\Gamma ^ \bot }^{\rm T} < 0 , {\Lambda ^ \bot }\Psi {\Lambda ^ \bot }^{\rm T} < 0 \end{align} $$

    引理 2[24].  假设$ \pmb \xi \in {\bf R}^n $, $ P = {P^{\rm T}} \in {\bf R}^{n \times n} $, $ H \in {\bf R}^{m \times n} $, rank$ (H) = r < n $, 则下列命题等价:

    i) $ {\pmb \xi ^{\rm T}}P\pmb \xi < 0, \; \; \forall H\pmb \xi = 0, \pmb \xi \ne 0 $

    ii) $ \exists X\in {\bf R}^{n \times m}, \; \; P + X^{\rm T}H^{\rm T} + H {X} < 0 $

    引理 3[25].  对于增广系统(3), 假设存在条件$ \det ({z_1}{z_2}I - $ $ {z_2}{\bar A_1}-{z_1}{\bar A_2})^{- 1}\ne 0 $, $ \forall \left({{z_1}, {z_2}} \right) \in \Big\{ {\left( {{z_1}, {z_2}} \right) \in {{\bf C} \times {\bf {C}}}:\left| {{z_1}} \right| \ge 1, \left| {{z_2}} \right| \ge 1} \Big\} $, 给定对称矩阵$ {\Theta} $和标量$ {\bar \omega _1}, $ $ {\bar \omega _2} \in \left[ {0, \pi } \right] $, 如果存在对称矩阵$ {P_k}, $ $ {Q_k}>0 \in{ {{\bf {C}}}^{n \times n}} $, $ k = 1, 2 $, 使得下式成立:

    $$ \begin{align} {\left[ {\begin{array}{*{20}{c}} {\bar A}&{{{\bar B}_f}}\\ I&0 \end{array}} \right]^{\rm T}}{\Sigma}\left[ {\begin{array}{*{20}{c}} {\bar A}&{{{\bar B}_f}}\\ I&0 \end{array}} \right] + {\Theta}< 0 \end{align} $$ (13)

    其中

    $$ \begin{equation} \begin{aligned} \bar A & = \left[ {{{\bar A}_1}, {{\bar A}_2}} \right], {\Sigma } = \left[ {\begin{array}{*{20}{c}} P&Q\\ *&\Delta \end{array}} \right]\\ P& = {P_1} + {P_2}, Q = \left[ {\begin{array}{*{20}{c}} {Q_1}&{Q_2} \end{array}} \right], {\bar B_f} = \left[ {\begin{array}{*{20}{c}} {\bar B_{f1}}&{\bar B_{f2}} \end{array}} \right]\\ \Delta& = {\rm diag}\left\{{-{P_1}-2\cos{{\bar\omega}_1}{Q_1}, -{P_2}- 2\cos{{\bar\omega}_2}{Q_2}} \right\}\\ \bar C & = {\rm diag}\left\{ {\tilde C, \tilde C} \right\}, {\bar D_f} = {\rm diag}\left\{ {{{\tilde D}_f}, {{\tilde D}_f}} \right\} \end{aligned} \end{equation} $$ (14)

    则下面的有限频条件成立:

    $$ \begin{equation} \left[ {\begin{array}{*{20}{c}} {\bar G\left( {{\omega _1}, {\omega _2}} \right)}\\ {I\left( {{\omega _1}, {\omega _2}} \right)} \end{array}} \right]^{\rm T} {\Theta} \left[ {\begin{array}{*{20}{c}} {\bar G\left( {{\omega _1}, {\omega _2}} \right)}\\ {I\left( {{\omega _1}, {\omega _2}} \right)} \end{array}} \right] < 0, \; \forall ({\omega _1}, {\omega _2}) \in \Omega \end{equation} $$ (15)

    其中

    $$ \begin{equation} \begin{aligned} &\bar G\left( {{\omega _1}, {\omega _2}} \right) = {\left[ {\begin{array}{*{20}{c}} {{{\rm e}^{{\rm j}{\omega _2}}}G\left( {{\omega _1}, {\omega _2}} \right)}\\ {{{\rm e}^{{\rm j}{\omega _1}}}G\left( {{\omega _1}, {\omega _2}} \right)} \end{array}} \right]}\\ &I\left( {{\omega _1}, {\omega _2}} \right) = \left[ {\begin{array}{*{20}{c}} {{{\rm e}^{{\rm j}{\omega _2}}}{I}}\\ {{{\rm e}^{{\rm j}{\omega _1}}}{I}} \end{array}} \right]\\ &G({\omega _1}, {\omega _2}) = ({{\rm e}^{{\rm j}({\omega _1} + {\omega _2})}}{I_n} - {{\rm e}^{{\rm j}{\omega _2}}}{{\bar A_1}}-\\ &\qquad \qquad {{\rm e}^{{\rm j}{\omega _1}}}{\bar A_2})^{ - 1}({{\rm e}^{{\rm j}{\omega _2}}}{\bar B_{f1}} + {{\rm e}^{{\rm j}{\omega _1}}}{\bar B_{f2}})\\ &\Omega = \left[ { - {{\bar \omega }_1}, {{\bar \omega }_1}} \right] \times \left[ { - {{\bar \omega }_2}, {{\bar \omega }_2}} \right] \end{aligned} \end{equation} $$ (16)

    引理 4[26].  若存在正定矩阵$ P_{s1}, P_{s2} $使得下式成立

    $$ \begin{equation} {\left[ {\begin{array}{*{20}{c}} {\bar A}\\ I \end{array}} \right]^{\rm T}}\left[ {\begin{array}{*{20}{c}} P_s&0\\ 0&{{\rm diag}\left\{ { - {P_{s1}}, - {P_{s2}} }\right\}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\bar A}\\ I \end{array}} \right] < 0 \end{equation} $$ (17)

    则增广系统(3)渐近稳定, 其中$ P_s = P_{s1}+P_{s2} $.

    定理 1.  给定标量$ {\bar \omega _{11}} $, $ {\bar \omega _{12}} \in \left[ {0, \pi } \right] $, $ {\gamma _1} > 0 $, $ \alpha > 0 $, 如果存在对称矩阵$ {P_{k1}} $, $ {P_{k3}} $, $ {Q_{k1}} $, $ {Q_{k3}} $和矩阵$ {P_{k2}} $, $ {Q_{k2}} $, $ k = 1, 2 $, $ {G_1} $, $ {G_2} $, $ {G_3} $, $ {F_1} $, $ {F_2} $, $ {F_3} $, $ {F_4} $, $ {H_1} $, $ {H_2} $, $ {\tilde A_1} $, $ {\tilde A_2} $, $ {\tilde B_1} $, $ {\tilde B_2} $, $ {\hat C_0} $, $ {\hat C_c} $使不等式(18)$ \, \sim\, $(20)成立, 则增广系统(3)满足性能指标(9).

    $$ \begin{equation} {\alpha ^2} - {\hat C_0}T < 0 \end{equation} $$ (18)
    $$ \begin{equation} \left[ {\begin{array}{*{20}{c}} {{Q_{k1}}}&{{Q_{k2}}}\\ {Q_{k2}^{\rm T}}&{{Q_{k3}}} \end{array}} \right] > 0, \quad k = 1, 2 \end{equation} $$ (19)
    $$ \begin{equation} \left[ {\begin{array}{*{20}{c}} {{\gamma _{11}}}&{{\gamma _{12}}}&{{\gamma _{13}}}&{{\gamma _{14}}}&{{\gamma _{15}}}&{{\gamma _{16}}}&{{\gamma _{17}}}&{{\gamma _{18}}}\\ *&{{\gamma _{22}}}&{{\gamma _{23}}}&{{\gamma _{24}}}&{{\gamma _{25}}}&{{\gamma _{26}}}&{{\gamma _{27}}}&{{\gamma _{28}}}\\ *&*&{{\gamma _{33}}}&{{\gamma _{34}}}&{{\gamma _{35}}}&{{\gamma _{36}}}&{{\gamma _{37}}}&{{\gamma _{38}}}\\ *&*&*&{{\gamma _{44}}}&{{\gamma _{45}}}&{{\gamma _{46}}}&{{\gamma _{47}}}&{{\gamma _{48}}}\\ *&*&*&*&{{\gamma _{55}}}&{{\gamma _{56}}}&{{\gamma _{57}}}&{{\gamma _{58}}}\\ *&*&*&*&*&{{\gamma _{66}}}&{{\gamma _{67}}}&{{\gamma _{68}}}\\ *&*&*&*&*&*&{{\gamma _{77}}}&{{\gamma _{78}}}\\ *&*&*&*&*&*&*&{{\gamma _{88}}} \end{array}} \right] < 0 \end{equation} $$ (20)

    其中

    $$ \begin{align*} {\gamma _{11}} = \, & {P_{11}}+{P_{21}} - {G_1} - G_1^{\rm T}\\ {\gamma _{12}} = \, & {P_{12}}+{P_{22}} - {G_3} - G_2^{\rm T}\\ {\gamma _{13}} = \, & {Q_{11}} + {G_1}{A_1} + {\tilde B_1}C- F_1^{\rm T}\\ {\gamma _{14}} = \, & {Q_{12}} + {G_1}{B_{u1}}{\hat C_c} + {\tilde A_1}- F_2^{\rm T}\\ {\gamma _{15}} = \, & {Q_{21}} + {G_1}{A_2} + {\tilde B_2}C- F_3^{\rm T}\nonumber\\ {\gamma _{16}} = \, & {Q_{22}} + {G_1}{B_{u2}}{\hat C_c} + {\tilde A_2}- F_4^{\rm T}\\ {\gamma _{17}} = \, & {G_1}{B_{f1}} + {\tilde B_1}{D_f} -H_1^{\rm T}\nonumber\\ {\gamma _{18}} = \, & {G_1}{B_{f2}} + {\tilde B_2}{D_f} -H_2^{\rm T}\nonumber\\ {\gamma _{22}} = \, & {P_{13}}+{P_{23}} - {G_3} - G_3^{\rm T}\\ {\gamma _{23}} = \, & Q_{12}^{\rm T} -G_{3}^{\rm T}+ {G_2}{A_1} + {\tilde B_1}C\nonumber\\ {\gamma _{24}} = \, & {Q_{13}}-{G_{3}^{\rm T}}+{G_2}{B_{u2}}{\hat C_c} +{\tilde A_1}\\ {\gamma _{25}} = \, & Q_{22}^{\rm T} -{G_{3}^{\rm T}}+ {G_2}{A_2} + {\tilde B_2}C\\ {\gamma_{26}} = \, & {G_2}{B_{u2}}{\hat C_c} + {\tilde A_2} + {Q_{23}}-G_3^{\rm T} \end{align*} $$
    $$ \begin{align} {\gamma _{27}} = \, & {G_2}{B_{f1}} + {\tilde B_1}{D_f}\\ {\gamma _{28}} = \, & {G_2}{B_{f2}} + {\tilde B_2}{D_f}\\ {\gamma_{33}} = \, & - {P_{11}} - 2\cos {\bar\omega _{11}}{Q_{11}} - {C^{\rm T}}C+\\ & {F_1}{A_1} + {\tilde B_1}C + {({F_1}{A_1} + {\tilde B_1}C)^{\rm T}}\\ {\gamma _{34}} = \, & - {P_{12}} - 2\cos {\bar\omega _{12}}{Q_{12}} - {C^{\rm T}}{\hat C_0}+\\ & ({F_1}{B_{u1}}{\hat C_c} + {\tilde A_1}) + {({F_2}{A_1} + {\tilde B_1}C)^{\rm T}}\\ {\gamma _{35}} = \, & {({F_3}{A_1} + {\tilde B_1}C)^{\rm T}} + {F_1}{A_2} + {\tilde B_2}C\\ {\gamma _{36}} = \, & {F_1}{B_{u2}}{\hat C_c} + {\tilde A_2} + {({F_4}{A_1} + {\tilde B_1}C)^{\rm T}} \\ {\gamma _{37}} = \, &{F_1}{B_{f1}} + {\tilde B_1}{D_f} + {({H_1}{A_1})^{\rm T}} -{{C}^{\rm T}}{D_f}\\ {\gamma _{38}} = \, &{F_1}{B_{f2}} + {\tilde B_2}{D_f} + {({H_2}{A_1})^{\rm T}}\\ {\gamma _{44}} = \, & - {P_{13}} - 2\cos {\bar\omega_{11}}{Q_{13}} +{F_2}{B_{u1}}{\hat C_c} + {\tilde A_1}+\\ & {({F_2}{B_{u1}}{\hat C_c} + {\tilde A_1})^{\rm T}} - {\alpha ^2}I \\ {\gamma _{45}} = \, & ({F_2}{A_2} + {\tilde B_2}C) + {({F_3}{B_{u1}}{\hat C_c} + {\tilde A_1})^{\rm T}}\\ {\gamma _{46}} = \, &{F_2}{B_{u2}}{\hat C_c} + {\tilde A_2} + {({F_4}{B_{u1}}{\hat C_c} + {\tilde A_1})^{\rm T}} \\ {\gamma _{47}} = \, & {F_2}{B_{f1}} + {\tilde B_1}{D_f} + {({H_1}{B_{u1}}{\hat C_c})^{\rm T}} + {{\hat C_0}^{\rm T}}{D_f} \\ {\gamma _{48}} = \, & {F_2}{B_{f2}} + {\tilde B_2}{D_f} + {({H_2}{B_{u1}}{\hat C_c})^{\rm T}} \\ {\gamma _{55}} = \, & - {P_{21}} - 2\cos {\bar\omega _{11}}{Q_{21}} + {F_3}{A_2} + {\tilde B_2}C +\\ & {({F_3}{A_2} + {\tilde B_2}C)^{\rm T}} - {C^{\rm T}}C \\ {\gamma _{56}} = \, & - {P_{22}} - 2\cos {\bar\omega _{12}}{Q_{22}} + {F_3}{B_{u2}}{\hat C_c}+ {\tilde A_2}+\\ & {C^{\rm T}}{\hat C_0} + {({F_4}{A_2} + {\tilde B_2}C)^{\rm T}} \\ {\gamma _{57}} = \, & {F_3}{B_{f1}} + {\tilde B_1}{D_f} + {({H_1}{A_2})^{\rm T}} \\ {\gamma _{58}} = \, & {F_3}{B_{f2}} + {({H_2}{A_2})^{\rm T}} + {\tilde B_2}{D_f} - {{C}^{\rm T}}{D_f}\\ {\gamma _{66}} = \, & - {P_{23}} - 2\cos {\bar\omega _{12}}{Q_{23}} - {{\alpha ^2}I } + {F_4}{B_{u2}}{\hat C_c}+\\ & {\tilde A_2} + {({F_4}{B_{u2}}{\hat C_c} + {\tilde A_2})^{\rm T}}, \\ {\gamma _{67}} = \, & {F_4}{B_{f1}} + {\tilde B_1}{D_f} + {({H_1}{B_{u2}}{\hat C_c})^{\rm T}}\\ {\gamma _{68}} = \, &{F_4}{B_{f2}} + {\tilde B_2}{D_f} + {({H_2}{B_{u2}}{\hat C_c})^{\rm T}} + {{\hat C_0}}{D_f} \\ {\gamma _{77}} = \, & {H_1}{B_{f1}} + {({H_1}{B_{f1}})^{\rm T}} + {\gamma _1}^2I - {D_f}^{\rm T}{D_f} \\ {\gamma _{78}} = \, & {H_1}{B_{f2}} + {({H_2}{B_{f1}})^{\rm T}}\\ {\gamma _{88}} = \, & {H_2}{B_{f2}} + {({H_2}{B_{f2}})^{\rm T}} + {\gamma _1}^2I - {D_f}^{\rm T}{D_f} \end{align} $$ (21)

    证明.  令式(13)中的

    $$ \begin{equation*} \begin{aligned} \Theta = \, &\left[ {\begin{array}{*{20}{c}} {\bar C}&{\bar D_f}\\ I&0 \end{array}} \right]^{\rm T}\left[ {\begin{array}{*{20}{c}} {-I}&0\\ 0&{\gamma _1}^2I \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\bar C}&{\bar D_f}\\ I&0 \end{array}} \right] \end{aligned} \end{equation*} $$

    则有限频条件(15)等价于$ {{G^{\rm T}}_{\tilde yf}}({\omega _1}, {\omega _2}){G}_{\tilde yf}({\omega _1}, {\omega _2})>{\gamma _1}^2I $, 即性能指标(9).由引理3知, 若式(13)成立, 则增广系统(3) $满足性能指标(9).式(13)可改写为

    $$ \begin{equation} \begin{aligned} {\Lambda^{\rm T}}\Omega \ {\Lambda } < 0 \end{aligned} \end{equation} $$ (22)

    其中

    $$ \begin{align*} \Lambda = \, & \left[ {\begin{array}{*{20}{c}} {{{\bar A}^{\rm T}}}&I&0\\ {\bar B_f^{\rm T}}&0&I \end{array}} \right]^{\rm T} \end{align*} $$
    $$ \begin{align*} \Omega = \, & \left[ {\begin{array}{*{20}{c}} I&0\\ 0&I\\ 0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} P&Q\\ *&\Delta \end{array}} \right]\left[ {\begin{array}{*{20}{c}} I&0&0\\ 0&I&0 \end{array}} \right]+\\ & \left[ {\begin{array}{*{20}{c}} 0&0\\ {{{\bar C}^{\rm T}}}&0\\ {\bar D_f^{\rm T}}&I \end{array}} \right]\left[ {\begin{array}{*{20}{c}} { - I}&0\\ *&{{\gamma _1}^2I} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 0&{\bar C}&{{\bar D_f}}\\ 0&0&I \end{array}} \right] \end{align*} $$

    $$ \begin{equation*} \Gamma = \left[ {\begin{array}{*{20}{c}} { - I}&{\bar A}&{{{\bar B}_f}} \end{array}} \right]^{\rm T}, \quad \pmb \eta = \Lambda \pmb \xi \end{equation*} $$

    则有$ {\Gamma ^{{\rm T}}}\pmb \eta = {\Gamma ^{{\rm T}}}\Lambda \pmb \xi = 0 $.根据引理2, 若以下不等式成立:

    $$ \begin{equation*} \Omega + \Gamma {\rm X} + {{\rm X}^{\rm T}}{\Gamma ^{\rm T}} < 0 \end{equation*} $$

    则式(22)成立.取$ {X} = \left[ {\begin{array}{*{20}{c}} {{G^{\rm T}}}&{{F^{\rm T}}}&{{H^{\rm T}}} \end{array}} \right] $,

    $$ \begin{equation*} \begin{aligned} G = \left[ {\begin{array}{*{20}{c}} {{G_1}}&{{G_3}}\\ {{G_2}}&{{G_3}} \end{array}} \right], F = \left[ {\begin{array}{*{20}{c}} {{F_1}}&{{G_3}}\\ {{F_2}}&{{G_3}}\\ {{F_3}}&{{G_3}}\\ {{F_4}}&{{G_3}} \end{array}} \right], H = \left[ {\begin{array}{*{20}{c}} {{H_1}}&0\\ {{H_2}}&0 \end{array}} \right] \end{aligned} \end{equation*} $$

    令$ {\tilde A_k} = {G_3}{\hat A_k} $, $ {\tilde B_k} = {G_3}{\hat B_k} $, $ k = 1, 2 $, 并将式(4)中相关矩阵代入, 可得

    $$ \begin{equation*} \left[ {\begin{array}{*{20}{c}} {{\gamma _{11}}}&{{\gamma _{12}}}&{{\gamma _{13}}}&{{\gamma _{14}}}&{{\gamma _{15}}}&{{\gamma _{16}}}&{{\gamma _{17}}}&{{\gamma _{18}}}\\ *&{{\gamma _{22}}}&{{\gamma _{23}}}&{{\gamma _{24}}}&{{\gamma _{25}}}&{{\gamma _{26}}}&{{\gamma _{27}}}&{{\gamma _{28}}}\\ *&*&{{\gamma _{33}}}&{{\gamma _{34}}}&{{\gamma _{35}}}&{{\gamma _{36}}}&{{\gamma _{37}}}&{{\gamma _{38}}}\\ *&*&*&{{{\tilde \gamma }_{44}}}&{{\gamma _{45}}}&{{\gamma _{46}}}&{{\gamma _{47}}}&{{\gamma _{48}}}\\ *&*&*&*&{{\gamma _{55}}}&{{\gamma _{56}}}&{{\gamma _{57}}}&{{\gamma _{58}}}\\ *&*&*&*&*&{{{\tilde \gamma }_{66}}}&{{\gamma _{67}}}&{{\gamma _{68}}}\\ *&*&*&*&*&*&{{\gamma _{77}}}&{{\gamma _{78}}}\\ *&*&*&*&*&*&*&{{\gamma _{88}}} \end{array}} \right] < 0 \end{equation*} $$

    其中

    $$ \begin{equation*} \begin{aligned} {\tilde \gamma _{44}} = \, & - {P_{13}} - 2\cos {\bar \omega _{11}}{Q_{13}} +{F_2}{B_{u1}}{\hat C_c} + {\tilde A_1}+\\ & {({F_2}{B_{u1}}{\hat C_c} + {\tilde A_1})^{\rm T}} - \hat C_0^{\rm T}{\hat C_0}\\ {\tilde \gamma _{66}} = \, & - {P_{23}} - 2\cos {\bar \omega _{12}}{Q_{23}} -{\hat C_0}^{\rm T}{\hat C_0} + {F_4}{B_{u2}}{\hat C_c}+\\ & {\tilde A_2} + {({F_4}{B_{u2}}{\hat C_c} + {\tilde A_2})^{\rm T}}\\ \end{aligned} \end{equation*} $$

    其他参数在式(21)中给出.需要注意的是$ {\tilde \gamma _{44}} $、$ {\tilde \gamma _{66}} $中存在耦合项$ \hat C_0^{\rm T}{\hat C_0} $.下面采用文献[27]中提出的方法, 给出处理耦合项的方案.假设$ {\hat C_0} $为行向量, 首先给出$ \hat C_0^{\rm T}{\hat C_0} $的上界, 即

    $$ \begin{equation} \hat C_0^{\rm T}{\hat C_0} > {\alpha ^2}I \end{equation} $$ (23)

    上式表明$ {\hat C_0} $的可行解是非凸的, 令

    $$ {\hat C_0}T - {\left\| T \right\|_2}^2 = 0 $$

    其中, $ {\left\| T \right\|_2} = \alpha $表示半径为$ \alpha $的球的切平面, 则通过约束条件(18)即可找到式(23)的解的凸子集.由引理3可知, 若式(18)$ \, \sim\, $(20)成立, 则增广系统(3)满足性能指标(9).

    注4.  该方法需要假设$ {\hat C_0} $为行向量, 即系统为单输入, 具有一定的局限性.

    接下来考虑系统故障检测的鲁棒性条件.令$ \pmb f(i, j) = 0 $, 则增广系统变为

    $$ \begin{align} \bar{\pmb x}(i + 1, j + 1) = \, &{\bar A_1}\pmb x(i, j + 1) + {\bar A_2} \pmb x(i + 1, j)+\\ & {\bar B_{d1}}\pmb d(i, j + 1) + {\bar B_{d2}}\pmb d(i + 1, j) \end{align} $$ (24)

    下面定理给出增广系统(3)满足性能指标(11)的充分条件.

    定理 2.   给定标量$ {\bar \omega_{21}} $, $ {\bar\omega_{22}}\in\left[{0, \pi}\right] $, $ {\gamma_2}>0 $, 如果存在对称矩阵$ {P_{m1}} $, $ {P_{m2}} $, $ {Q_{m1}>0} $, $ {Q_{m2}}>0 $, 矩阵$ {M_1} $, $ {M_2} $, $ {G_3} $, $ {\tilde A_1} $, $ {\tilde A_2} $, $ {\tilde B_1} $, $ {\tilde B_2} $, $ {\hat C_0} $, $ {\hat C_c} $使不等式(25)成立, 则增广系统(3)满足性能指标(11).

    $$ \begin{equation*} \left[ {\begin{array}{*{20}{c}} {{\Gamma _{11}}}&{{\Gamma _{12}}}&{{\Gamma _{13}}}&0\\ *&{{\Delta _m}}&0&{{\Gamma _{24}}}\\ *&*&{ - {\gamma _2}^2I}&{{\Gamma _{34}}}\\ *&*&*&{ - I} \end{array}} \right] < 0 \end{equation*} $$ (25)

    其中

    $$ \begin{equation*} \begin{aligned} \Gamma _{11} = \, & {P_{m1}} + {P_{m2}} - He\left[ {\begin{array}{*{20}{c}} {{M_1}}&{{G_3}}\\ {{M_2}}&{{G_3}} \end{array}} \right]\\ {\Gamma _{12}} = \, & \left[ {\begin{array}{*{20}{c}} {{Q_{m1}}}&{{Q_{m2}}} \end{array}} \right]+\left[ {\begin{array}{*{20}{c}} {{\Gamma_{m1}}}&{{\Gamma_{m2}}} \end{array}} \right]\\ {{\Gamma_{m1}}} = \, &\left[\begin{array}{cc} {{M_1}{A_1} + {{\tilde B}_1}C}&{{M_1}{{ B}_{u1}}{{\hat C}_c} + {{\tilde A}_1}}\\ {{M_2}{A_1} + {{\tilde B}_1}C}&{{M_2}{{ B}_{u1}}{{\hat C}_c} + {{\tilde A}_1}}\\ \end{array}\right]\\ {{\Gamma_{m2}}} = \, &\left[\begin{array}{cc} {{M_1}{A_2} + {{\tilde B}_2}C}&{{M_1}{{ B}_{u2}}{{\hat C}_c} + {{\tilde A}_2}}\\ {{M_2}{A_2} + {{\tilde B}_2}C}&{{M_2}{{ B}_{u2}}{{\hat C}_c} + {{\tilde A}_2}}\\ \end{array}\right]\\ \Gamma _{13} = \, &\left[ {\begin{array}{*{20}{c}} {{M_1}{B_{d1}} + {{\tilde B}_1}{D_d}}&{{M_1}{B_{d2}} + {{\tilde B}_2}{D_d}}\\ {{M_2}{B_{d1}} + {{\tilde B}_1}{D_d}}&{{M_2}{B_{d2}} + {{\tilde B}_2}{D_d}} \end{array}} \right]\\ {\Gamma _{24}} = \, &\left[ {\begin{array}{*{20}{c}} {{C^{\rm T}}}&0\\ { - {{\hat C}_0}^{\rm T}}&0\\ 0&{{C^{\rm T}}}\\ 0&{ - {{\hat C}_0}^{\rm T}} \end{array}} \right], {\Gamma _{34}} = {\rm diag}\left\{ {{{\tilde D}_d}, {{\tilde D}_d}} \right\} \end{aligned} \end{equation*} $$

    证明.  由引理3知, 若存在矩阵

    $$ \begin{align*} {P_m} = \, & {P_{m1}} + {P_{m2}}, {Q_m} = \left[ {\begin{array}{*{20}{c}} {{Q_{m1}}}&{{Q_{m2}}}\end{array}}\right], \\ {\Delta _m} = \, & {\rm diag}\Big\{ - {P_{m1}} - 2\cos {{\bar \omega }_{21}}{Q_{m1}}, - {P_{m2}} - \\& 2\cos {{\bar \omega }_{22}}{Q_{m2}} \Big\} \end{align*} $$

    使如下不等式成立:

    $$ \begin{equation*} \begin{aligned} &\left[ {\begin{array}{*{20}{c}} {\bar A}&{\bar B_d}\\ I&0 \end{array}} \right]^{\rm T}\left[ {\begin{array}{*{20}{c}} P_m&Q_m\\ *&\Delta_m \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\bar A}&{\bar B_d}\\ I&0 \end{array}} \right]+\\ & \qquad \left[ {\begin{array}{*{20}{c}} {\bar C}&{\bar D_d}\\ I&0 \end{array}} \right]^{\rm T}\left[ {\begin{array}{*{20}{c}} { I}&0\\ 0&{{-\gamma _2}^2I} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\bar C}&{\bar D_d}\\ I&0 \end{array}} \right]<0 \end{aligned} \end{equation*} $$

    则增广系统(3)满足性能指标(11).上式可改写为

    $$ \begin{equation} {{\Upsilon} ^{\rm T}}{\Omega_1} {\Upsilon } < 0 \end{equation} $$ (26)

    其中

    $$ \begin{equation*} \begin{aligned} \Upsilon = \, & \left[ {\begin{array}{*{20}{c}} {{{\bar A}^{\rm T}}}&I&0\\ {\bar B_d^{\rm T}}&0&I \end{array}} \right]^{\rm T}\\ {\Omega_1} = \, & \left[ {\begin{array}{*{20}{c}} I&0\\ 0&I\\ 0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} P_m&Q_m\\ *&\Delta_m \end{array}} \right]\left[ {\begin{array}{*{20}{c}} I&0&0\\ 0&I&0 \end{array}} \right]+\\ & \left[ {\begin{array}{*{20}{c}} 0&0\\ {{{\bar C}^{\rm T}}}&0\\ {\bar D_d^{\rm T}}&I \end{array}} \right]\left[ {\begin{array}{*{20}{c}} { I}&0\\ *&{{-\gamma _2}^2I} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 0&{\bar C}&{{\bar D_d}}\\ 0&0&I \end{array}} \right]\\ {\bar B_d} = \, & \left[ {\begin{array}{*{20}{c}} {{{\bar B}_{d1}}}&{{{\bar B}_{d2}}} \end{array}} \right]\\ {\bar D_d} = \, & \left[ {\begin{array}{*{20}{c}} {{{\tilde D}_d}}&0\\ 0&{{{\tilde D}_d}} \end{array}} \right] \end{aligned} \end{equation*} $$

    $$ \begin{equation*} \Gamma_m = {\left[ {\begin{array}{*{20}{c}} { - I}&{\bar A}&{{{\bar B}_d}} \end{array}} \right]}^{\rm T} \end{equation*} $$

    则$ {{\Gamma_m}^ \bot } = \left[ {\begin{array}{*{20}{c}} {{{\bar A}^{\rm T}}}&I&0\\ {\bar B_d^{\rm T}}&0&I \end{array}} \right] $.根据引理1, 下式与式(26)等价,

    $$ \begin{equation*} {\Omega_1}+ He\left( {\Gamma_m {M^{\rm T}}{Z ^{\rm T}}} \right) < 0 \end{equation*} $$ (27)

    取$ Z = \left[ {\begin{array}{*{20}{c}} I&0&0 \end{array}} \right]^{\rm T} $, $ M = \left[ {\begin{array}{*{20}{c}} {{M_1}}&{{G_3}}\\ {{M_2}}&{{G_3}} \end{array}} \right] $, 并令$ {\tilde A_k} = {G_3}{\hat A_k} $, $ {\tilde B_k} = {G_3}{\hat B_k} $, $ k = 1, 2 $, 利用Schur $补引理可证.

    定理 3.  给定标量$ {\bar \omega_{11}} $, $ {\bar\omega_{12}}\in\left[{0, \pi}\right] $, $ {\beta _1} > 0 $, 如果存在对称矩阵$ {P_{r1}} $, $ {P_{r2}} $, $ {Q_{r1}>0} $, $ {Q_{r2}>0} $, 矩阵$ {R_1} $, $ {R_2} $, $ {G_3} $, $ {\tilde A_1} $, $ {\tilde A_2} $, $ {\tilde B_1} $, $ {\tilde B_2} $, $ {\hat C_0} $, $ {\hat C_c} $使得不等式(26)成立, 则增广系统(3)满足性能指标(10).

    $$ \begin{equation} \left[ {\begin{array}{*{20}{c}} {{\Xi _{11}}}&{{\Xi _{12}}}&{{\Xi _{13}}}&0\\ *&{{\Delta _r}}&0&{{\Xi _{24}}}\\ *&*&{ - \beta_1^2I}&0\\ *&*&*&{ - I} \end{array}} \right] < 0 \end{equation} $$ (28)

    其中

    $$ \begin{equation*} \begin{aligned} {\Xi _{11}} = \, & {P_{r1}} + {P_{r2}} - He\left[ {\begin{array}{*{20}{c}} {{R_1}}&{{G_3}}\\ {{R_2}}&{{G_3}} \end{array}} \right]\\ {\Xi _{12}} = \, & \left[ {\begin{array}{*{20}{c}} {{Q_{r1}}}&{{Q_{r2}}} \end{array}} \right]+ \left[ {\begin{array}{*{20}{c}} {{\Xi_{r1}}}&{{\Xi_{r2}}} \end{array}} \right]\\ {{\Xi_{r1}}} = \, &\left[\begin{array}{cc} {{R_1}{A_1} + {{\tilde B}_1}C}&{{R_1}{{ B}_{u1}}{{\hat C}_c} + {{\tilde A}_1}}\\ {{R_2}{A_1} + {{\tilde B}_1}C}&{{R_2}{{ B}_{u1}}{{\hat C}_c} + {{\tilde A}_1}}\\ \end{array}\right]\\ {{\Xi_{r2}}} = \, &\left[\begin{array}{cc} {{R_1}{A_2} + {{\tilde B}_2}C}&{{R_1}{{ B}_{u2}}{{\hat C}_c} + {{\tilde A}_2}}\\ {{R_2}{A_2} + {{\tilde B}_2}C}&{{R_2}{{ B}_{u2}}{{\hat C}_c} + {{\tilde A}_2}}\\ \end{array}\right] \nonumber\\ {\Xi _{13}} = \, &\left[ {\begin{array}{*{20}{c}} {{R_1}{B_{f1}} + {{\tilde B}_1}{D_f}}&{{R_1}{B_{f2}} + {{\tilde B}_2}{D_f}}\\ {{R_2}{B_{f1}} + {{\tilde B}_1}{D_f}}&{{R_2}{B_{f2}} + {{\tilde B}_2}{D_f}} \end{array}} \right]\\ {\Xi _{24}} = \, & \left[ {\begin{array}{*{20}{c}} {{E^{\rm T}}}&0\\ { ({F_u}\hat C_c)^{\rm T}}&0\\ 0&{{E^{\rm T}}}\\ 0&{ ({F_u}\hat C_c)^{\rm T}} \end{array}} \right]\end{aligned} \end{equation*} $$
    $$ \begin{equation*} {\Delta _r} = {\rm diag}\{ { - {P_{r1}} - 2\cos {{\bar \omega }_{11}}{Q_{r1}}, - {P_{r2}} - 2\cos {{\bar \omega }_{12}}{Q_{r2}}} \} \end{equation*} $$

    证明.  参考定理2证明过程.

    定理 4.  给定标量$ {\bar \omega_{21}} $, $ {\bar\omega_{22}}\in\left[{0, \pi}\right] $, $ {\beta _2} > 0 $, 如果存在对称矩阵$ {P_{u1}} $, $ {P_{u2}} $, $ {Q_{u1}>0} $, $ {Q_{u2}}>0 $, 矩阵$ {U_1} $, $ {U_2} $, $ {G_3} $, $ {\tilde A_1} $, $ {\tilde A_2} $, $ {\tilde B_1} $, $ {\tilde B_2} $, $ {\hat C_0} $, $ {\hat C_c} $使得不等式(27)成立, 则增广系统$ (3)满足性能指标(12).

    $$ \begin{equation} \left[ {\begin{array}{*{20}{c}} {{\Omega _{11}}}&{{\Omega _{12}}}&{{\Omega _{13}}}&0\\ *&{{\Delta _u}}&0&{{\Xi _{24}}}\\ *&*&{ - \beta _2^2I}&0\\ *&*&*&{ - I} \end{array}} \right] < 0 \end{equation} $$ (29)

    其中

    $$ \begin{equation*} \begin{aligned} {\Omega _{11}} = \, & {P_{u1}} + {P_{u2}} - He\left[ {\begin{array}{*{20}{c}} {{U_1}}&{{G_3}}\\ {{U_2}}&{{G_3}} \end{array}} \right]\\ {\Omega _{12}} = \, &\left[ {\begin{array}{*{20}{c}} {{Q_{u1}}}&{{Q_{u2}}} \end{array}} \right]+\left[ {\begin{array}{*{20}{c}} {\Omega _{u1}}&{\Omega _{u2}} \end{array}} \right]\\ {\Omega _{u1}} = \, &\left[\begin{array}{cc} {{U_1}{A_1} + {{\tilde B}_1}C}&{{U_1}{{ B}_{u1}}{{\hat C}_c} + {{\tilde A}_1}}\\ {{U_2}{A_1} + {{\tilde B}_1}C}&{{U_2}{{ B}_{u1}}{{\hat C}_c} + {{\tilde A}_1}}\\ \end{array}\right]\\ {\Omega _{u2}} = \, &\left[\begin{array}{cc} {{U_1}{A_2} + {{\tilde B}_2}C}&{{U_1}{{ B}_{u2}}{{\hat C}_c} + {{\tilde A}_2}}\\ {{U_2}{A_2} + {{\tilde B}_2}C}&{{U_2}{{ B}_{u2}}{{\hat C}_c} + {{\tilde A}_2}}\\ \end{array}\right] \nonumber\\ {\Omega _{13}} = \, & \left[ {\begin{array}{*{20}{c}} {{U_1}{B_{d1}} + {{\tilde B}_1}{D_d}}&{{U_1}{B_{d2}} + {{\tilde B}_2}{D_d}}\\ {{U_2}{B_{d1}} + {{\tilde B}_1}{D_d}}&{{U_2}{B_{d2}} + {{\tilde B}_2}{D_d}} \end{array}} \right] \end{aligned} \end{equation*} $$
    $$ \begin{equation*} {\Delta _u} = {\rm diag}\{ { - {P_{u1}} - 2\cos {{\bar \omega }_{21}}{Q_{u1}}, - {P_{u2}} - 2\cos {{\bar \omega }_{22}}{Q_{u2}}} \} \end{equation*} $$

    证明.  参考定理2证明过程.

    定理1~4给出了故障检测滤波器/控制器设计需满足的有限频域性能条件, 由于广义KYP引理并不隐含系统的稳定性[28], 因此这些条件并不能保证所设计的系统是稳定的.下面给出增广系统(3)渐近稳定的充分条件:

    定理 5.  如果存在正定矩阵$ { P_{s1}} $, $ { P_{s2}} $, 矩阵$ {S_1} $, $ {S_2} $, $ {G_3} $, $ {\tilde A_1} $, $ {\tilde A_2} $, $ {\tilde B_1} $, $ {\tilde B_2} $使不等式(28)成立, 则增广系统(3)渐近稳定.

    $$ \begin{equation} \left[ {\begin{array}{*{20}{c}} {{\Psi _{11}}}&{{\Psi _{12}}}\\ *&{{\Psi _{22}}} \end{array}} \right] < 0 \end{equation} $$ (30)

    其中

    $$ \begin{equation*} \begin{aligned} {\Psi _{11}} = \, & { P_{s1}} + { P_{s2}} - \left[ {\begin{array}{*{20}{c}} {{S_1} + S_1^{\rm T}}&{{G_3} + S_2^{\rm T}}\\ {G_3^{\rm T} + {S_2}}&{G_3^{\rm T} + {G_3}} \end{array}} \right]\\ {\Psi _{12}} = \, & \left[ {\begin{array}{*{20}{c}} \Psi _{s1}&\Psi _{s2} \end{array}} \right]\\ \Psi _{s1} = \, & \left[\begin{array}{cc} {{S_1}{A_1} + {{\tilde B}_1}C}&{{{\tilde A}_1}+{S_1}{B_{u1}}{\hat C_c}}\\ {{S_2}{A_1} + {{\tilde B}_1}C}&{{{\tilde A}_1}+{S_2}{B_{u1}}{\hat C_c}}\\ \end{array}\right]\\ \Psi _{s2} = \, & \left[\begin{array}{cc} {{S_1}{A_2} + {{\tilde B}_2}C}&{{{\tilde A}_2}+{S_1}{B_{u2}}{\hat C_c}}\\ {{S_2}{A_2} + {{\tilde B}_2}C}&{{{\tilde A}_2}+{S_2}{B_{u2}}{\hat C_c}}\\ \end{array}\right] \nonumber\\ {\Psi _{22}} = \, & {\rm diag}\{ { - {{ P}_{s1}}, - {{ P}_{s2}}} \}\end{aligned} \end{equation*} $$

    证明.  由引理4及引理1易证.

    上述定理中得到的矩阵不等式为非凸的, 为了解决该难题, 采取两步算法进行求解:

    步骤1.  设计状态反馈控制器, 使闭环系统(32)满足控制性能指标(10)和(12), 得到控制器参数$ {\hat C_c} $.

    设计如下形式的状态反馈控制器:

    $$ \begin{equation} \pmb u(i, j) = {\hat C_c}\pmb x(i, j) \end{equation} $$ (31)

    可得闭环系统:

    $$ \begin{align} \pmb x(i + 1, j + 1) = \, &({A_1} + {B_{u1}}{{\hat C}_c})\pmb x(i, j + 1) +\\ & ({A_2} + {B_{u2}}{{\hat C}_c})\pmb x(i + 1, j)+\\ & {B_{d1}}\pmb d(i, j + 1) + {B_{d2}}\pmb d(i + 1, j)+\\ & {B_{f1}}\pmb f(i, j + 1) + {B_{f2}}\pmb f(i + 1, j)\\ \pmb z(i, j) = \, &(E + {F_u}{{\hat C}_c})\pmb x(i, j)+{F_f}\pmb f(i , j) \end{align} $$ (32)

    定义矩阵

    $$ \begin{equation*} \begin{aligned} \hat A = \, & \left[ {\begin{array}{*{20}{c}} {{A_1} + {B_{u1}}{{\hat C}_c}}&{{A_2} + {B_{u2}}{{\hat C}_c}} \end{array}} \right] = \left[ {{\hat A_{1}}, {\hat A_{2}}} \right]\\ F_f = \, &0, \hat F_f = \left[ {\begin{array}{*{20}{c}} {F_f}&0\\ 0&{F_f} \end{array}} \right] \\\hat E = \, & \left[ {\begin{array}{*{20}{c}} {E + {F_u}{{\hat C}_c}}&0\\ 0&{E + {F_u}{{\hat C}_c}} \end{array}} \right], {\hat B_f} = \left[ {{B_{f1}}, {B_{f2}}} \right]\end{aligned} \end{equation*} $$

    定理 6.  给定标量$ {\beta _1} > 0 $, $ {\bar \omega _{11}}, {\bar \omega _{12}} \in \left[ {0, \pi } \right] $, 如果存在对称矩阵$ {\bar P_1} $, $ {\bar P_2} $及$ {\bar Q_1}>0 $, $ {\bar Q_2}>0 $, 矩阵$ {X} $, $ {Y} $, $ {V_{i}} $, $ i = 1, \cdots , 7 $, 使不等式(31)成立, 则系统(32) $满足性能指标(10).此外, 若式(31)成立, 则状态反馈控制器$ {\hat C_c} = YX^{-1} $.

    $$ \begin{equation} \left[ {\begin{array}{*{20}{c}} {{\Phi _{11}}}&{{\Phi _{12}}}&{{\Phi _{13}}}&{{\Phi _{14}}}&{{\Phi _{15}}}&{{\Phi _{16}}}&{{\Phi _{17}}}\\ *&{{\Phi _{22}}}&{{\Phi _{23}}}&{{\Phi _{24}}}&{{\Phi _{25}}}&{{\Phi _{26}}}&{{\Phi _{27}}}\\ *&*&{{\Phi _{33}}}&{{\Phi _{34}}}&{{\Phi _{35}}}&{{\Phi _{36}}}&{{\Phi _{37}}}\\ *&*&*&{{\Phi _{44}}}&{{\Phi _{45}}}&{{\Phi _{46}}}&{{\Phi _{47}}}\\ *&*&*&*&{{\Phi _{55}}}&{{\Phi _{56}}}&{{\Phi _{57}}}\\ *&*&*&*&*&{{\Phi _{66}}}&{{\Phi _{67}}}\\ *&*&*&*&*&*&{{\Phi _{77}}} \end{array}} \right] < 0 \end{equation} $$ (33)

    其中

    $$ \begin{align*} {\Phi _{11}} = \, & {\bar P_1} + {\bar P_2}- {X} - X^{\rm T} , {\Phi _{12}} = {\bar Q_1}\\ {\Phi _{13}} = \, & - V_{1}^{\rm T}, {\Phi _{14}} = 0\\ {\Phi _{15}} = \, & {({A_1}{X} +{B_{f1}}{V_1}+{B_{u1}}{Y})^{\rm T}}+{\bar {Q_2}} \\ {\Phi _{16}} = \, &{({EX + {F_u}Y})^{\rm T}}, \Phi_{17} = 0, \\ {\Phi _{22}} = \, & -\bar P_1-2\cos{\bar \omega_{11}} \bar Q_1-X-X^{\rm T}\\ \Phi_{23} = \, &0, {\Phi _{24}} = - V_{5}^{\rm T}\\ {\Phi _{25}} = \, & {({A_2}{X} + {B_{f2}}{V_5} +{B_{u2}}{Y})^{\rm T}}, \Phi_{26} = 0\\ {\Phi _{27}} = \, & {({EX + {F_u}Y})^{\rm T}}, {\Phi _{33}} = I- {V_{2}} - {V_{2}}^{\rm T} \end{align*} $$
    $$ \begin{align*} \Phi_{34} = \, &0, {\Phi _{35}} = -{V_{3}} + (B_{f1}V_2)^{\rm T}, \Phi_{36} = 0 \\ {\Phi _{37}} = \, & - {V_{4}}, {\Phi _{44}} = I- {V_{6}} - {V_{6}}^{\rm T}\\ {\Phi _{45}} = \, &{({B_{f2}}{V_{6}})^{\rm T}}, {\Phi _{46}} = - {V_{7}} , \Phi_{47} = 0\\ {\Phi _{55}} = \, &-\bar{P_2}-2\cos\bar{\omega}_{12}\bar{Q}_2+B_{f1}V_3+(B_{f1}V_3)^{\rm T}\\ {\Phi _{56}} = \, & {B_{f2}}{V_{7}}, {\Phi _{57}} = {B_{f1}}{V_{4}}, {\Phi _{66}} = - \beta _1^2I\\ \Phi_{67} = \, &0, {\Phi _{77}} = - \beta _1^2I \end{align*} $$

    证明.  根据广义KYP引理的对偶形式, 若下式

    $$ \begin{align} &\left[ {\begin{array}{*{20}{c}} {\hat A}&{I}\\ {\hat E}&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} \bar P&\bar Q\\ *&\bar \Delta \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\hat A}&{I}\\ {\hat E}&0 \end{array}} \right]^{\rm T}+\\ & \qquad \left[ {\begin{array}{*{20}{c}} {\hat B_f}&{0}\\ {\hat F_f}&I \end{array}} \right]\left[ {\begin{array}{*{20}{c}} { I}&0\\ 0&{{-\beta _1}^2I} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\hat B_f}&{0}\\ {\hat F_f}&I \end{array}} \right]^{\rm T}<0 \end{align} $$ (34)

    成立, 则系统(32)满足性能指标(10).式(34)可写为

    $$ \begin{equation} \left[ {\begin{array}{*{20}{c}} S&I \end{array}} \right] K \left[ {\begin{array}{*{20}{c}} {{\Sigma _1}}&0\\ 0&{{\Pi _1}} \end{array}} \right]{K ^{\rm T}}{\left[ {\begin{array}{*{20}{c}} S&I \end{array}} \right]^{\rm T}} < 0 \end{equation} $$ (35)

    其中

    $$ \begin{equation*} \begin{aligned} K = \, &\left[{\begin{array}{*{20}{c}} I&0&0&0&0&0&0\\ 0&I&0&0&0&0&0\\ 0&0&0&I&0&0&0\\ 0&0&0&0&I&0&0\\ 0&0&I&0&0&0&0\\ 0&0&0&0&0&I&0\\ 0&0&0&0&0&0&I \end{array}} \right]\\ \Sigma_1 = \, &\left[ {\begin{array}{*{20}{c}} \bar P&\bar Q\\ *&\bar \Delta \end{array}} \right], \Pi_1 = \left[ {\begin{array}{*{20}{c}} { I}&0\\ 0&{{-\beta _1}^2I} \end{array}} \right]\\ S = \, &\left[ {\begin{array}{*{20}{c}} {\hat A}&{{{\hat B}_f}}\\ {\hat E}&{0} \end{array}} \right] = \\ \, & \left[ {\begin{array}{*{20}{c}} {A_1}&{A_{2}}&{B_{f1}}&{B_f}\\E&0&0&0\\0&E&0&0 \end{array}} \right]+\\ &\left[ {\begin{array}{*{20}{c}} {B_{u1}}&{B_{u2}}\\{F_{u}}&{0}\\0&{F_u} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} {\hat C_c}&{0}\\{0}&{\hat C_c} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} I&0&0&0\\0&I&0&0 \end{array}} \right] = \\ \, &\mathscr{A}+\mathscr{B}L\mathscr{C}\end{aligned} \end{equation*} $$

    $$ \begin{equation*} \begin{aligned} {X_1} = {\{{\mathscr{C}}^\dagger XR+(I-{\mathscr{C}^ \dagger}\mathscr{C})V\} } , \quad Y = {\hat C_c}X \end{aligned} \end{equation*} $$

    $$ \begin{equation*} \begin{aligned} SX_1 = \mathscr{A}X_1+\mathscr{B}YR \end{aligned} \end{equation*} $$

    由于$ \mathscr{C} = \left[ {\begin{array}{*{20}{c}} I&0&0&0\\0&I&0&0 \end{array}} \right] $, 故$ \mathscr{C}^\dagger = \left[ {\begin{array}{*{20}{c}} I&0\\0&I\\0&0\\0&0 \end{array}} \right] $, 因此可得

    $$ \begin{equation*} \begin{aligned} {X_1} = \, &\left[ {\begin{array}{*{20}{c}} {XR_1}\\ {XR_2} \\{ V_a}\\{ V_b} \end{array}} \right] \end{aligned} \end{equation*} $$

    令$ {P_1} = K \left[ {\begin{array}{*{20}{c}} {{\Sigma _1}}&0\\ 0&{{\Pi _1}} \end{array}} \right]{K ^{\rm T}}, H_1 = \begin{bmatrix} -I\\S \end{bmatrix}, $ $ H_1^ \bot = \left[ {\begin{array}{*{20}{c}} S&I \end{array}} \right] $, 式(35)可改写为

    $$ \begin{equation*} {P_1} + He({H_1}{X_1}) < 0 \end{equation*} $$

    $$ \begin{equation*} \begin{aligned} {R_1} = \, & \left[ {\begin{array}{*{20}{c}} I&0&0&0&0&0&0 \end{array}} \right]\\ {R_2} = \, &\left[ {\begin{array}{*{20}{c}} 0&I&0&0&0&0&0 \end{array}} \right]\\ V_a = \, &\begin{bmatrix} V_1&0&V_2&0&V_3&0&V_4 \end{bmatrix}\\ V_b = \, &\begin{bmatrix} 0&V_5&0&V_6&0&V_7&0 \end{bmatrix} \end{aligned} \end{equation*} $$

    代入可得式(33).此外, 若式(33)成立, 易知$ X $可逆, 则状态反馈控制器$ {\hat C_c} = YX^{-1} $.

    定理 7.  给定标量$ {\beta _2} > 0 $, $ {\bar \omega _{21}}, {\bar \omega _{22}} \in \left[ {0, \pi } \right] $, 如果存在对称矩阵$ {\tilde P_1} $, $ {\tilde P_2} $及$ {\tilde Q_1}>0 $, $ {\tilde Q_2}>0 $, 矩阵$ {X} $, $ {Y} $, $ {V_{i}} $, $ i = 1, \cdots , 7 $, 使不等式(36)成立, 则系统(32) $满足性能指标(12).此外, 若式(36)成立, 则状态反馈控制器$ {\hat C_c} = YX^{-1} $.

    $$ \begin{equation} \left[ {\begin{array}{*{20}{c}} {{\Sigma _{11}}}&{{\Sigma _{12}}}&{{\Phi _{13}}}&{{\Sigma _{14}}}&{{\Sigma _{15}}}&{{\Phi _{16}}}&{{\Phi _{17}}}\\ *&{{\Sigma _{22}}}&{{\Phi _{23}}}&{{\Phi _{24}}}&{{\Sigma _{25}}}&{{\Phi _{26}}}&{{\Phi _{27}}}\\ *&*&{{\Phi _{33}}}&{{\Phi_{34}}}&{{\Sigma _{35}}}&{{\Phi _{36}}}&{{\Phi _{37}}}\\ *&*&*&{{\Sigma _{44}}}&{{\Sigma _{45}}}&{{\Phi _{46}}}&{{\Phi _{47}}}\\ *&*&*&*&{{\Sigma _{55}}}&{{\Sigma _{56}}}&{{\Sigma _{57}}}\\ *&*&*&*&*&{{\Sigma _{66}}}&{{\Phi _{67}}}\\ *&*&*&*&*&*&{{\Sigma _{77}}} \end{array}} \right] < 0 \end{equation} $$ (36)

    其中

    $$ \begin{equation*} \begin{aligned} {\Sigma _{11}} = \, & {\tilde P_1} + {\tilde P_2}- {X} - X^{\rm T} , {\Sigma _{12}} = {\tilde Q_1}\\ {\Sigma _{14}} = \, &0 \\ {\Sigma _{15}} = \, & {({A_1}{X} +{B_{d1}}{V_1}+{B_{u1}}{Y})^{\rm T}}+{\tilde {Q_2}}\\ {\Sigma _{22}} = \, & -{\tilde P_1} -2\cos {\bar \omega _{21}}{\tilde Q_1}- {X} - X^{\rm T} \\ {\Sigma _{25}} = \, & {({A_2}{X} + {B_{d2}}{V_5} +{B_{u2}}{Y})^{\rm T}}\\ {\Sigma _{35}} = \, & -{V_{3}} + ({B_{d1}}{V_2})^{\rm T}\\ {\Sigma _{44}} = \, & I- {V_{6}} - {V_{6}}^{\rm T}\\ {\Sigma _{45}} = \, &{({B_{d2}}{V_{6}})^{\rm T}} \\ {\Sigma _{55}} = \, &-\tilde{P_2}-2\cos\bar{\omega}_{22}\tilde{Q}_2+B_{d1}V_3+(B_{d1}V_3)^{\rm T} \\ {\Sigma _{56}} = \, & {B_{d2}}{V_{7}}, {\Sigma _{57}} = {B_{d1}}{V_{4}}, {\Sigma _{66}} = - \beta _2^2I, {\Sigma _{77}} = - \beta _2^2I \end{aligned} \end{equation*} $$

    证明.  参考定理6证明过程.

    通过求解以下优化问题获得控制器参数$ {\hat C_c} $:

    $$ \begin{equation*} \begin{aligned} &\min a{\beta _1} + b{\beta _2}\\ &{\rm s.t.}\;\;(33), (36) \end{aligned} \end{equation*} $$ (37)

    给定实参数$ a $, $ b $, 若上述优化问题可解, 则控制器参数$ {\hat C_c} = YX^{-1} $.

    步骤2.  在步骤1的基础上($ {\hat C_c} $已知), 给定实参数$ {a_1} $, $ {a_2} $, $ {b_1} $, $ {b_2} $, 求解如下的优化问题:

    $$ \begin{equation*} \begin{aligned} & \min {a_2}{\gamma _2} + {b_1}{\beta _1} + {b_2}{\beta _2} - {a_1}{\gamma _1}\\ &{\rm s.t.}\;(18), (19), (20), (25), (28), (29), (30) \end{aligned} \end{equation*} $$

    若上述优化问题可解, 则可得滤波器参数$ \hat A_k = G_3^{-1}\tilde A_k, \hat B_k = G_3^{-1}\tilde B_k, k = 1, 2, {\hat C_0} = {\hat C_0} $.

    注5.  与一维系统相比, 二维系统的稳定性条件和性能条件复杂, 其故障检测滤波器/控制器的设计过程更加困难.求解有限频$ {H_ - } $指标时, 一维系统可通过Finsler定理等避免出现非凸问题, 而二维系统由于其广义KYP引理的特殊形式, 需要通过构造切平面方法来解决设计过程中出现的非凸问题.二维系统的状态反馈控制器设计难度也更大, 需利用广义KYP引理的对偶形式进行构造性证明.

    受参考文献[29]启发, 选择如下残差评价函数及阈值:

    $$ \begin{equation} \begin{aligned} {J_r}(i, j)& = \sqrt{\frac{{\sum\limits_{p = 0}^s {\sum\limits_{q = 0}^t {{r^{\rm T}}}(i-p, j-q)r(i-p, j-q)}}}{{(s+1)(t+1)}}}\\ {J_{th}}& = \mathop{\sup}\limits_{f = 0, d \ne 0} {J_r}(i, j) \end{aligned} \end{equation} $$ (38)

    其中, $ {J_r}(i, j) $和$ {J_{th}} $分别表示残差函数及阈值.阈值$ {J_{th}} $可借鉴文献[30]中的算法求出.根据如下的逻辑关系检测系统是否发生了故障:

    $$ \begin{array}{l} J\left( {i,j} \right) > {J_{th}} \Rightarrow 系统存在故障 \Rightarrow 报警\\ J\left( {i,j} \right) < {J_{th}} \Rightarrow 系统无故障 \Rightarrow 不报警 \end{array} $$

    注6.  对于二维系统而言, 残差评价函数需要从水平和垂直两个方向定义以反映二维特性.本文选择水平方向$ i-s $到$ i $, 垂直方向$ j-t $$到$ j $的矩形区域内残差的平均值作为评价函数.而对于一维系统, 残差评价函数只需要在一个方向上定义, 其评估窗口通常为$ k_0 $到$ k_0+n $的时间范围.

    考虑文献[15]中给出的带钢轧制过程, 如图 1所示,

    图 1  带钢轧制过程
    Fig. 1  Strip rolling process

    该轧制过程可用如下等式描述:

    $$ \begin{equation*} \begin{aligned} \pmb y_{i}(t) = {\frac{\lambda}{\lambda +Mp^2}}\left\{\left(1+\frac{Mp^2}{\lambda_1}\right)\pmb y_{i-1}(t)-{\frac{1}{\lambda_2}}\pmb F_m\right\} \end{aligned} \end{equation*} $$

    其中, $ p $表示微分算子$ {\rm d}/{\rm d}t $, $ \pmb y_i(t) $是第$ i $个实际压辊间隙厚度, $ \pmb F_m $是由电动机产生的力, $ M $是压辊间隙调节机构的总质量, $ \lambda_1 $表示调节机构弹簧, $ \lambda_2 $表示带钢的硬度, $ \lambda = \lambda_1\lambda_2/(\lambda_1+\lambda_2) $是带钢和压辊机构的复合刚度.通过向后差分并用$ T_1 $表示采样周期, 则上式可以用离散时间形式表示:

    $$ \begin{equation*} \begin{aligned} \pmb y_i(t+T_1) = \, & c_1\pmb y_i(t)+c_2\pmb y_i(t-T_1)+c_3\pmb y_{i-1}(t+T_1)+\\ & c_4\pmb y_{i-1}(t)+c_5\pmb y_{i-1}(t-T_1)+b \pmb u_{i}(t)\end{aligned} \end{equation*} $$

    $$ \begin{equation*} \begin{aligned} c_1 = \, &\frac{2M}{\lambda {T_1}^2+M}, \; c_2 = \frac{-M}{\lambda {T_1}^2+M}\\ c_3 = \, & \frac{\lambda}{\lambda {T_1}^2+M}\left({T_1}^2+\frac{M}{\lambda }\right)\\ c_4 = \, & -\frac{2\lambda M}{\lambda_1(\lambda{T_1}^2+M)}, \; c_5 = \frac{\lambda M}{\lambda_1(\lambda{T_1}^2+M)}\\ b = \, & -\frac{\lambda {T_1}^2}{\lambda_2(\lambda{T_1}^2+M)}\\ \pmb x(i, j): = \, &[\pmb y_{i-1}^{\rm T}((j+1)T_1)\quad \pmb y_{i-1}^{\rm T}(jT_1)\quad \pmb y_{i}^{\rm T}(jT_1)\\ &\pmb y_{i}^{\rm T}((j-1)T_1)\quad \pmb y_{i-1}^{\rm T}((j-1)T_1)]^{\rm T}\\ \pmb u(i, j): = \, &\pmb F_m, \; \pmb y(i, j): = \pmb y_i(jT_1) \end{aligned} \end{equation*} $$

    上述等式可转化为FM模型:

    $$ \begin{equation*} \begin{split} {\pmb x}({ i} + 1, { j} + 1) = \, &A_1{\pmb x}({ i}, { j} + 1) + A_2{\pmb x}({ i} + 1, { j})+\\ & B_{{ d}1}{\pmb d}({ i}, { j} + 1) + B_{{ d}2}{\pmb d}({ i} + 1, { j})+\\ & B_{{ u}1}{\pmb u}({ i}, j + 1) + B_{u2}{\pmb u}({ i}+1, j) +\\ & B_{{ f}1}{\pmb f}({ i}, j + 1) + B_{{ f}2}{\pmb f}({ i} + 1, { j})\\ \pmb y({ i}, { j}) = \, & C{\pmb x}({ i}, { j}) + D_{ d}{\pmb d}({ i}, { j}) + D_{ f}{\pmb f}({ i}, { j})\\ \pmb z({ i}, { j}) = \, & E{\pmb x}({ i}, { j}) + F_{ u} {\pmb u}({ i}, { j}) \end{split} \end{equation*} $$

    其中

    $$ \begin{align*} {A_1}& = \left[ {\begin{array}{*{20}{c}} {c_3}&{c_4}&{c_1}&{c_2}&{c_5}\\ 0&0&1&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0 \end{array}} \right]\\{A_2}& = \left[ {\begin{array}{*{20}{c}} 0&0&0&0&0\\0&0&0&0&0\\{c_3}&{c_4}&{c_1}&{c_2}&{c_5}\\0&0&1&0&0\\0&1&0&0&0 \end{array}} \right]\\ {B_{d1}}& = \left[ {\begin{array}{*{20}{c}} 0.0605\\0.3993\\0.5269\\0.4168\\0.6569 \end{array}} \right], {B_{d2}} = \left[ {\begin{array}{*{20}{c}} 0.6280\\0.2920\\0.4317\\0.0155\\0.9841 \end{array}} \right] \end{align*} $$
    $$ \begin{align*} {B_{u1}}& = \left[ {\begin{array}{*{20}{c}} b\\0\\0\\0\\0 \end{array}} \right], {B_{u2}} = \left[ {\begin{array}{*{20}{c}} 0\\0\\b\\0\\0 \end{array}} \right]\\ B_{f1}& = \begin{bmatrix} -1.0414&0&0&0&0 \end{bmatrix}^{\rm T}\\B_{f2}& = \begin{bmatrix} 0&0&-1.0414&0&0 \end{bmatrix}^{\rm T}\\ C& = \begin{bmatrix} 0&0&0&0&0 \end{bmatrix}\\ E& = \left[ {\begin{array}{*{20}{c}} 0&0&{1}&0&0 \end{array}} \right]\\ F_u& = 0.1672, D_f = 1.2, D_d = 0.5 \end{align*} $$

    令$ \lambda_1 = \lambda_2 = 1\, 800, T_1 = 0.8, M = 100 $, 假设上述系统中发生了卡死型传感器故障.利用本文所提出的方法设计故障检测滤波器/控制器, 同时保证一定的故障检测性能和控制性能.给定加权值$ {a_1} = 0.1 $, $ {a_2} = 0.4 $, $ {b_1} = 0.4 $, $ {b_2} = 0.1 $, 由于故障发生在低频段, 取频率约束$ {\bar \omega _{k1}} = {\bar \omega _{k2}} = \pi /12, k = 1, 2 $, 其余参数取为$ T = \left[ {\begin{array}{*{20}{c}} 0.8147&0.9058&0.1270&0.9134&0.6324 \end{array}} \right]^{\rm T} $, $ \alpha = 1.6537 $.根据上节所提出的算法, 可得如下故障检测滤波器/控制器参数:

    $$ \begin{equation*} \begin{aligned} {\hat A_1} = \, & \left[ {\begin{array}{*{20}{c}} -0.0673&-0.0120&0.0238&-0.0189&0.0103\\ -0.0064&-0.1977&0.0372&0.0114&0.0109\\ 0.0065&0.0130&-0.1643&0.0184&0.0082\\ -0.0037&0.0061&0.0135&-0.1931&0.0030\\ -0.0196&0.0166&0.0128&0.0091&-0.1854\\ \end{array}} \right]\\ {\hat A_2} = \, & \left[ {\begin{array}{*{20}{c}} -0.1618&0&0.0071&-0.0030&0.0026\\ 0.0097&-0.1998&0.0160&0.0041&0.0088\\ 0.0601&0.0040&-0.1529 & 0 & 0.0137\\ 0.0099&0.0047&0.0384&-0.2012&0.0050\\ -0.0148&0.0316&0.0103&0.0077&-0.1892 \end{array}} \right]\\ {\hat B_1} = \, & \left[ {\begin{array}{*{20}{c}} 0.3024\\ -0.0514\\ 0.0357\\ -0.0827\\ -0.0437 \end{array}} \right], {\hat B_2} = \left[ {\begin{array}{*{20}{c}} -0.1311\\ 0.0634\\ 0.2529\\ 0.0884\\ -0.0510 \end{array}} \right] \end{aligned} \end{equation*} $$
    $$ \begin{equation*} \begin{aligned} {\hat C_c} = \, & \left[ {\begin{array}{*{20}{c}} 556.2519&-81.8249&86.8303&-88.8581&44.4291 \end{array}} \right]\\ {\hat C_0} = \, &\left[ {\begin{array}{*{20}{c}} 1.0215&1.1350&0.1381&1.1477&0.7935 \end{array}} \right] \end{aligned} \end{equation*} $$

    为了验证该故障检测滤波器/控制器的有效性, 给出仿真结果(图 1~4).在仿真中, 考虑卡死型故障

    图 2  故障$ \pmb f(i, j)$
    Fig. 2  Fault $ \pmb f(i, j)$
    图 3  干扰$ \pmb d(i, j)$
    Fig. 3  Disturbance $ \pmb d(i, j)$
    图 4  三维空间中的残差评价函数$ {J_r}(i, j)$和阈值$ {J_{th}}$
    Fig. 4  Residual evalution function $ {J_r}(i, j)$ and threshold $ {J_{th}}$ in three-dimensional space
    $$ \begin{equation*} \pmb f(i, j) = \begin{cases} 0.8, &40 \le i \le 50, 40 \le j \le 150\\ 0, & \mbox{否则} \end{cases} \end{equation*} $$

    以及干扰

    $$ \begin{equation*} \pmb d(i, j) = 0.15{\sin}(i){\rm e}^{-0.02i}+0.3{\cos}(j){\rm e}^{-0.03j} \end{equation*} $$

    系统初始状态设为$ {{\pmb x}_i}(k, 1) = 0 $, $ {\hat {\pmb x}_i}(k, 1) = 0 $, $ {\pmb x_i}(1, k) = 0 $, $ {\hat {\pmb x}_i}(1, k) = 0 $.采用参考文献[30]中的算法, 可得阈值$ {J_{th}} = 0.8251 $. 图 2表示系统的故障, 图 3表示系统干扰, 图 4图 5分别给出了三维空间和二维空间的故障检测效果.从图 5可以看出, 当$ i = 15, 23, 35 $时, 残差评价函数的值位于阈值的下方, 表明系统没有发生故障; 当$ i = 44 $, $ 40 \le j\le150 $时, 残差评价函数的值位于阈值的上方, 表明此时系统发生了故障, 这与预设的故障一致, 因此该设计可以有效地检测出故障发生.

    图 5  二维空间中的残差评价函数$ {J_r}(i, j)$和阈值$ {J_{th}}$
    Fig. 5  Residual evalution function $ {J_r}(i, j)$ and threshold $ {J_{th}}$ in two-dimensional space

    接下来与传统的分步设计方法[15]进行比较, 仍采用上文中的轧制模型中的数据, 先设计控制器, 再设计故障检测滤波器.在相同条件下进行仿真研究, 仿真结果如图 6~8所示. 图 6为故障到残差传递函数的奇异值比较, 可以看出集成设计方法的故障灵敏性更高(奇异值更大). 图 7图 8为两种方法的控制性能(即扰动抑制性能)的比较, 可以看出当故障发生时, 本文所提出的集成设计方法具有更好的扰动抑制能力.

    图 6  故障到残差传递函数${G_{\tilde yf}}({\omega _1}, {\omega _2})$的奇异值比较
    Fig. 6  Singular value comparison of transfer functions ${G_{\tilde yf}}({\omega _1}, {\omega _2})$ from fault to residual signal
    图 7  集成设计方法的控制性能
    Fig. 7  performance of integrated design methods
    图 8  分步设计方法的控制性能
    Fig. 8  Control performance of separate design methods

    本文研究了FM模型的同时故障检测与控制问题.借助于二维广义KYP引理, 直接处理系统需要满足的有限频性能指标, 可避免频率加权方法的复杂性.所设计的故障检测滤波器/控制器, 在检测故障发生的同时, 还可以满足给定的控制性能指标.利用构造切平面方法以及两步法来解决设计过程中出现的非凸问题.仿真例子验证了方法的有效性.


  • 本文责任编委 张俊
  • 图  1  平行系统基本框架

    Fig.  1  The basic framework of parallel system

    图  2  突发事件分类

    Fig.  2  Category of emergency

    图  3  突发事件应对流程图

    Fig.  3  Flow chart of treatment for emergency

    图  4  基于ACP方法的平行应急疏散系统框架设计思路

    Fig.  4  The design idea of ACP-based parallel emergency evacuation systems

    图  5  集成平台的框架结构

    Fig.  5  Framework of integration platform

    图  6  基于代理的人工应急疏散系统

    Fig.  6  Agent-based artificial parallel emergency evacuation systems

    图  7  平行应急疏散系统计算实验框架

    Fig.  7  Framework of computational experiments in parallel emergency evacuation systems

    图  8  平行执行框架及原理

    Fig.  8  Framework and principle of parallel execution

    图  9  轨道交通枢纽站平行应急疏散系统平台:硬件配置(左)和软件平台界面(右)

    Fig.  9  Platform of parallel emergency evacuation systems (PeES) for rail transportation hub stations: hardware configurations (left) and the interface of software platform (right)

    图  10  某枢纽站2D平面图

    Fig.  10  The 2D floor plan of a hub station

    图  11  疏散开始30 s, 45 s和60 s时2种不同方案下站台乘客疏散快照

    Fig.  11  The snapshots of passenger evacuation under different evacuation strategies at the time of 30 s, 45 s and 60 s

    图  12  站台剩余乘客数与疏散时间的关系

    Fig.  12  Number of passengers in the subway platform against evacuation time under different evacuation strategies

    表  1  人员Agent模型默认属性值

    Table  1  Default values of individual agent's characteristics

    参数属性默认值
    $m_i$质量80 kg
    $r_i$半径0.3 m
    $\tau_i$弛豫时间0.5 s
    $A_i$回避力强度2 000 N
    $B_i$回避因子0.08 m
    $\kappa$滑动摩擦因子 $2.4 \times 10^5 {\rm kgm}^{-1} {\rm s}^{-1}$
    $k$身体压缩因子 $1.2 \times 10^5 {\rm kgm}^{-1}s^{-1}$
    $v_i^0$期望速度N$(1.34, 0.26) {\rm ms}^{-1}$
    $R_{inf}$影响半径11 m
    $R_{vf}$视野半径10 m
    $\beta_i$愿意跟随程度1
    $a_i$引导信息可得度1
    $b_1$位置反馈因子0.05
    $b_2$速度反馈因子0.05
    $\lambda _i$各向异性0.5
    下载: 导出CSV

    表  2  乘客应急疏散计算实验结果(s)

    Table  2  Experimental results of passenger emergency evacuation (s)

    对照方案方案1方案2
    单次疏散时间1468781
    平均疏散时间143.292.580.8
    标准差28.48.25.5
    下载: 导出CSV
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